next | previous | forward | backward | up | top | index | toc | Macaulay2 web site
Macaulay2Doc :: solve

solve -- solve a linear equation

Synopsis

Description

(Disambiguation: for division of matrices, which can also be thought of as solving a system of linear equations, see instead Matrix // Matrix. For lifting a map between modules to a map between their free resolutions, see extend.)

There are several restrictions. The first is that there are only a limited number of rings for which this function is implemented. Second, over RR or CC, the matrix A must be a square non-singular matrix. Third, if A and b are mutable matrices over RR or CC, they must be dense matrices.
i1 : kk = ZZ/101;
i2 : A = matrix"1,2,3,4;1,3,6,10;19,7,11,13" ** kk

o2 = | 1  2 3  4  |
     | 1  3 6  10 |
     | 19 7 11 13 |

              3        4
o2 : Matrix kk  <--- kk
i3 : b = matrix"1;1;1" ** kk

o3 = | 1 |
     | 1 |
     | 1 |

              3        1
o3 : Matrix kk  <--- kk
i4 : x = solve(A,b)

o4 = | 2  |
     | -1 |
     | 34 |
     | 0  |

              4        1
o4 : Matrix kk  <--- kk
i5 : A*x-b

o5 = 0

              3        1
o5 : Matrix kk  <--- kk
Over RR or CC, the matrix A must be a non-singular square matrix.
i6 : printingPrecision = 2;
i7 : A = matrix "1,2,3;1,3,6;19,7,11" ** RR

o7 = | 1  2 3  |
     | 1  3 6  |
     | 19 7 11 |

                3          3
o7 : Matrix RR    <--- RR
              53         53
i8 : b = matrix "1;1;1" ** RR

o8 = | 1 |
     | 1 |
     | 1 |

                3          1
o8 : Matrix RR    <--- RR
              53         53
i9 : x = solve(A,b)

o9 = | -.15 |
     | 1.1  |
     | -.38 |

                3          1
o9 : Matrix RR    <--- RR
              53         53
i10 : A*x-b

o10 = | 0        |
      | -3.3e-16 |
      | -8.9e-16 |

                 3          1
o10 : Matrix RR    <--- RR
               53         53
i11 : norm oo

o11 = 8.88178419700125e-16

o11 : RR (of precision 53)
For large dense matrices over RR or CC, this function calls the lapack routines.
i12 : n = 10;
i13 : A = random(CC^n,CC^n)

o13 = | .18+.32i .87+.15i .63+.14i   .73+.68i  .59+.18i .27+.84i .53+.89i
      | .67+.43i .12+.89i .75+.05i   .48+.51i  .82+.95i .39+.52i .56+.41i
      | .43+.18i .49+.15i .051+.081i .64+.21i  .39+.6i  .33+.37i .78+.29i
      | .76+.01i .7+.55i  .62+.1i    .088+.48i .42+.2i  .56+.07i .89+.26i
      | .94+.88i .74+.51i .67+.62i   .91+.55i  .27+.63i .42+.67i .007+.2i
      | .5+.11i  .85+.04i .61+.98i   .86+.52i  .5+.68i  1+.61i   .48+.12i
      | .35+.93i .97+.56i .85+.62i   .23+.78i  .36+.12i .19+.95i .56+.05i
      | .99+.47i .33+.16i .46+.64i   .42+.95i  .49+.14i .74+.78i .3+.68i 
      | .97+.04i .91+.94i .78+.19i   .13+.075i .87+.74i .45+.44i .82+.17i
      | .42+.61i .13+.19i .05+.8i    .87+.23i  .52+.39i .48+.51i .87+.94i
      -----------------------------------------------------------------------
      .45+.21i  .15+.77i  .75+.52i  |
      .11+.6i   .34+.41i  .4+.41i   |
      .32+.47i  .44+.93i  .18+.35i  |
      .2+.54i   .37+.71i  .46+.52i  |
      1+.32i    .008+.28i .3+.13i   |
      .25+.54i  .26+.43i  .32+.2i   |
      .77+.19i  .57+.99i  .72+.33i  |
      .095+.18i .96+.88i  .61+.7i   |
      .14+.95i  .24+.28i  .046+.11i |
      .84+.74i  .19+.67i  .73+.25i  |

                 10          10
o13 : Matrix CC     <--- CC
               53          53
i14 : b = random(CC^n,CC^2)

o14 = | .9+.48i    .18+.9i  |
      | .26+.27i   .45+.99i |
      | .04+.54i   .55+.4i  |
      | .052+.026i .46+.86i |
      | .41+.22i   .96+.53i |
      | .68+.73i   .91+.74i |
      | .97+.53i   .37+.45i |
      | .88+.24i   .64+.95i |
      | .56+.52i   .25+.55i |
      | .89+.24i   .1+.8i   |

                 10          2
o14 : Matrix CC     <--- CC
               53          53
i15 : x = solve(A,b)

o15 = | -.21-.097i 1.7+.56i  |
      | .54i       -1.2+.37i |
      | .54-.2i    .62-.26i  |
      | -.4+.28i   2.2+.3i   |
      | .39+.32i   -.04+1.6i |
      | .08-.69i   -1.3-1.1i |
      | -.26-.48i  .19+2.5i  |
      | .23-.53i   -1.1-.94i |
      | .44+.11i   -1.5-.77i |
      | .097+.45i  .58-1.3i  |

                 10          2
o15 : Matrix CC     <--- CC
               53          53
i16 : norm ( matrix A * matrix x - matrix b )

o16 = 1.27675647831893e-15

o16 : RR (of precision 53)
This may be used to invert a matrix over ZZ/p, RR or QQ.
i17 : A = random(RR^5, RR^5)

o17 = | .31  .47 .12 .23 .038 |
      | .16  .97 .83 .93 .59  |
      | .067 .84 .76 .74 .23  |
      | .084 .51 .14 .88 .36  |
      | .83  .37 .85 .29 .3   |

                 5          5
o17 : Matrix RR    <--- RR
               53         53
i18 : I = id_(target A)

o18 = | 1 0 0 0 0 |
      | 0 1 0 0 0 |
      | 0 0 1 0 0 |
      | 0 0 0 1 0 |
      | 0 0 0 0 1 |

                 5          5
o18 : Matrix RR    <--- RR
               53         53
i19 : A' = solve(A,I)

o19 = | 1    -.72 -.54 .64  .94  |
      | 2.4  1.3  -.46 -1.3 -.97 |
      | -1.6 -.43 1.5  -.56 .6   |
      | -1.2 -2.1 1.5  2.2  .47  |
      | -.16 3.7  -3.5 -.79 -.26 |

                 5          5
o19 : Matrix RR    <--- RR
               53         53
i20 : norm(A*A' - I)

o20 = 4.16333634234434e-16

o20 : RR (of precision 53)
i21 : norm(A'*A - I)

o21 = 3.46944695195361e-16

o21 : RR (of precision 53)
Another method, which isn't generally as fast, and isn't as stable over RR or CC, is to lift the matrix b along the matrix A (see Matrix // Matrix).
i22 : A'' = I // A

o22 = | 1    -.72 -.54 .64  .94  |
      | 2.4  1.3  -.46 -1.3 -.97 |
      | -1.6 -.43 1.5  -.56 .6   |
      | -1.2 -2.1 1.5  2.2  .47  |
      | -.16 3.7  -3.5 -.79 -.26 |

                 5          5
o22 : Matrix RR    <--- RR
               53         53
i23 : norm(A' - A'')

o23 = 0

o23 : RR (of precision 53)

Caveat

This function is limited in scope, but is sometimes useful for very large matrices

See also

Ways to use solve :